A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

learn more… | top users | synonyms

0
votes
0answers
9 views

Is the space of all adapted processes with Càdlàg paths a Banach space?

Consider first the definition of a stochastic integral for simple predictable processes. $$I:\mathbb{S}\rightarrow\mathbb{D},\ H\mapsto I_X(H):=H_0X_0+\sum_{i=1}^nH_i(X_{T_{i+1}}-X_{T_i})$$ The ...
3
votes
0answers
43 views

A Markov Chain probability, conditioned on a random time.

My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set $A$ (if it matters you can set $A=\lbrace ...
1
vote
1answer
34 views

Existence stochastic integral

I am trying to understand the prove of the existence of the stochastic integral for a local martinglale null at $0$ and continuous, $M\in \mathcal{M}^c_{0,\text{loc}}$, a predictable process $H\in ...
1
vote
1answer
13 views

Relationship between minimizing a conditional variance and a covariance

We are working with discrete-time stochastic processes. Let $v_k$ be a $\mathcal F_k$-predictable process, and let $X_k, \eta_k$ be $\mathcal F_k$-adapted processes. Define $V_k = v_kX_k+\eta_k$ and ...
-1
votes
1answer
50 views

Show that $\mathbb{P}(\tau_{0}>T)\approx\frac{1}{\sqrt{T}}$ where $\{ B(t) : t\geq 0\}$ is a linear brownian motion started at $B(0)=1$ [on hold]

I'd appreciate if someone could provide me with a solution for the following problem: Let $\left\{ B\left(t\right)\thinspace|\thinspace t\geq0\right\}$ be a linear brownian motion started at ...
4
votes
2answers
58 views

How to take into account uncertainty on number of events

Suppose I generate a set of events $X_{i}$ for $i = 1,2 \dots N$ and suppose every event is either a success or a failure, ie. $X_{i} = 0, 1$. If $N$ is fixed, the MLE for the probability of success ...
1
vote
0answers
35 views

Normalized hit times of a simple RW converge in distribution to hit times of standard Brownian Motion

I would appreciate some hints or guidance towards solving the following exercise: Let $\left\{ S\left(j\right)\thinspace:\thinspace j=0,1,...\right\}$ be a simple random walk on the integers ...
0
votes
0answers
7 views

How to simulate Permanental Point Process

I have simulated a determinantal point process in a square grid using Gaussian Kernel. The Gaussain matrix is decomposed into its eigenvectors and eigenvalues. In core implementation, the elementary ...
1
vote
0answers
14 views

Fundamental theorem for Malliavin derivative and Lebesgue integral

I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics. Let ...
3
votes
5answers
120 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
1
vote
1answer
18 views

What is the probability of no events in a Markov-modulated Poisson process?

Suppose I have a two-state continuous-time Markov chain $M$ with rate matrix $Q$. $$ Q = \begin{bmatrix} -q_{01} & q_{01} \\ q_{10} & -q_{10} \end{bmatrix} $$ Now consider a Poisson process ...
4
votes
1answer
32 views

Brownian motion: Strong Markov versus translation invariance

In the proof of the reflection principle in Durrett's textbook (Probability: Theory and Examples (4e), Theorem 8.4.1, page 317), there's a step which I'm a little shaky on. Basically, this proof ...
0
votes
0answers
10 views

Gaussian Process with explicit basis functions

I am considering the Gaussian process with explicit basis functions as discussed in the book (section 2.7): http://www.gaussianprocess.org/gpml/chapters/RW2.pdf Has anyone tried to derive formulas ...
1
vote
1answer
15 views

Martingale and independent increment

I know that in $L^2$ martingale a have independent increments. In particular that $\mathbb{E}[(X_m-X_n)^2]=\mathbb{E}[X^2_m-X^2_n]$ if X is a martingale. Does this extend also for general $p\geq 1$ in ...
1
vote
0answers
19 views

Simple Stratonovich product for physical system

I was reading a physical textbook and they used the "Stratonovich product" referred to $v_1 \circ dW_1 = \frac{1}{2}[v_1 + (v_1+dv_1)]dW_1$. I think this product is from the Stochastic process, thus ...
1
vote
0answers
14 views

Distribution of hitting time for two border brownian motion

I'm trying to find the distribution of hitting times for two border brownian motion with respect to both the hitting time AND which border is hit. Is this well defined? This is assuming $W_0=0$ with ...
1
vote
0answers
15 views

Smith's Key Renewal Theorem for Renewal Function

Consider a renewal process $(N_t)_{t \geq 0}$ and its renewal function $M(t):=\mathbb{E}[N_t]$ with interarrival distribution function $F$. One can show that $M$ satisfies the $(F,F)$-renewal ...
0
votes
1answer
70 views

Analytic solution to stochastic differential equations

I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. ...
2
votes
0answers
20 views

Tail field versus germ field of Brownian motion

Continuing my foray into Brownian motion (apologies for the bombardment...), I'm trying to verify the details of a proof of Durrett of the following 0-1 property of the tail $\sigma$-algebra of ...
7
votes
1answer
130 views

When does $\sum_{i=1}^{\infty} X_i$ exist for random sequences $\{X_i\}_{i=1}^{\infty}$?

Suppose $\{X_1, X_2, X_3, \ldots\}$ is an infinite sequence of random variables such that $E[X_i]=0$ for all $i$, and $E[X_iX_j]=0$ whenever $i \neq j$. Further suppose the variances $\sigma_i^2 = ...
0
votes
1answer
17 views

Change from stochastic exponential to exponential of Lévy process - Applebaum

In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process. I am not sure about ...
0
votes
0answers
13 views

Formula for running-time complexity

I'm regarding a stochastic process $(X_t)$of which the mean starts at $O(n)$ and is reduced by the factor $(1-r)$ in each step with $r = \Omega (1/n^9)$, so $$E(X_{t+1}) \leq E(X_t) (1-r) .$$ Now it ...
2
votes
1answer
36 views

Independent increments of a Poisson process

In the following $\{X_t\}$ is a Poisson process. Assume that I've proved that $P(X_s=i,X_t-X_s=k)=P(X_s=i)P(X_t-X_s=k)$ so that the two events are independent, does it follow that ...
2
votes
1answer
27 views

Intuition about Blumenthal's 0-1 law

I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$. Let $\mathcal{F}_t^+ = \cap_{s ...
1
vote
0answers
26 views

Pure death processes

If $P_n (t)=\Pr (N (t)=n)$ and $N (0)=a$, how can I show that in a pure death process $$P_{(a-1)}(t)=a (e^{\mu t }-1)e^{-a \mu t}.$$ I showed that $P_a(t)=e^{-a \mu t}$. In fact I want to show ...
0
votes
1answer
28 views

Probability of time between two events in a poisson process

Suppose people arrive at a certain place according to a poisson process with rate 10 per day. 1) What is the expected time until the arrival of 100 person. 2) What is the probability that ...
3
votes
1answer
64 views

How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$ where $a\neq 0$ and $b\geq 1/2$, is ...
0
votes
0answers
5 views

2 dimensional stochastic partial differential equation [closed]

I am trying to code 2 dimensional stochastic heat equation ($u_t=u_{xx}+u_{yy}+dW(t,x,y)$) in matlab. But i am in confusion with $dW(t,x,y)$. Please suggest me that, how to code the stochastic term.
0
votes
1answer
27 views

What are some applications of stochastic processes and advances probabilities in real world? [closed]

One obvious field is Finance what are some others applications ?
0
votes
1answer
18 views

Closed communicating class

Let $P_{ij}$ a transition matrix, a class $C$ is closed if given two different states $i$ and $j$ $$i\in C, i\rightarrow j\Rightarrow j\in C$$ If a Markov Chain is irreducible the transition matrix ...
0
votes
2answers
62 views

Probability in a fixed die

I have that transition matrix is ...
0
votes
1answer
38 views

Why is $f(X_t)-\int_0^t Af(X_s) \, ds$ a martingale for a Markov process $(X_t)_{t \geq 0}$?

I think if $A$ is the usual generator for the Markov process $(X_t)_t$ $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}$$ then we get that for any "nice" $f$ the process ...
1
vote
0answers
20 views

differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)} w.r.t$ t?

Here $\phi(u,t)=E\{e^{iut}\} $ is a characteristic function, $x_t$ is Gaussian. Differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)}$ w.r.t t the result is $\phi_u=\phi[i\hat{x}_t-P_tu]$
1
vote
1answer
46 views

Understanding the Markov property of Brownian motion

I'm trying to understand the Markov property for Brownian motions in full generality. The textbook I'm following states it like this: Recall that we have a family of measures $P_x, x \in ...
0
votes
0answers
36 views

expected value of expected value

I want to quantify the error of phase noise in terms of its normalized mean squared error. I define the error measure as (x is the error free function, y the distorted): $$ \rm NMSE = \frac{\int ...
1
vote
0answers
73 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
3
votes
0answers
26 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
2
votes
1answer
22 views

Compute the expected value of a brownian motion

Suppose $X(t)$ is a brownian motion. Compute $E[X(1)X(5)X(7)]$. I know that the brownian motion has independent increments, so if we could write $X(1)X(5)X(7)$ as such, then we could use the ...
2
votes
2answers
76 views

Probability returning to initial state

Let $P=\begin{bmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&0\end{bmatrix}$ and $P^{(n+1)}=P^{(n)}P.$ I know that if you start in any ...
2
votes
0answers
22 views

Expected response time of Continuous time Markov chain

I'm studying CTMC (Continuous Time Markov Chains). I came across the following slide I don't understand how they got $M(t+h) = M(t) + \alpha h + M(t)\lambda h - M(t) \mu h +o(h)$ Could anyone ...
2
votes
0answers
12 views

Stationary distribution for Markov process with non-exponential waiting times

Stochastic processes often are described in terms of transition rates where the length of time waited before a transition occurs is an exponential random variable. For example: $0\rightarrow 1$ at ...
1
vote
1answer
22 views

Conditional probability and disjoint events

If $\cup_{n=1}^\infty B_n=\Omega$ and $P(\Omega)=1$ then $\sum_{n=1}^\infty P(B_n)=1$, now $$P(A)=\sum_{n=1}^\infty P(A|B_n)P(B_n)=p\sum_{i=1}^\infty P(B_n)=p$$ If $X$ and $Y$ are independents ...
0
votes
0answers
42 views

How to calculate hitting probabilities for Brownian motion.

Given a standard Brownian motion with no drift, the PDF is... $${{1} \over {t^{3/2} \cdot \sqrt{2\cdot \pi}}} \cdot e^{-1/{2t}}$$ (Derived from the CDF $\int_{-\infty}^{f(t)/\sqrt{t}} {1 \over {2 ...
3
votes
3answers
48 views

Find $p_{ij}^{(n)}$ for the transition matrix

Let $$P=\begin{bmatrix}\frac{1}{3}&0&\frac{2}{3}\\\frac{1}{3}&\frac{2}{3}&0\\\frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}$$ find ...
1
vote
1answer
28 views

What is a martingale array - its definition and importance?

What is a martingale array? What is the importance of defining such an array, instead of using a martingale itself? A common example of this definition is a martingale difference array.
0
votes
2answers
33 views

Markovian systems: Why must controls be independent of state?

I am currently working my way through Probabilistic Robotics by Thrun, Burgard, and Fox. On p. 91, I encountered the following statement: The Markovian assumption implies independence between ...
0
votes
0answers
8 views

Generating Correlated Samples: Cholesky Decomposition of Correlation Matrix or Covariance Matrix? [duplicate]

I have multiple correlated stochastic processes and I would like to generate correlated samples of them. From my understanding, if I have my samples $Z$ and a Cholesky decomposition of their ...
-1
votes
1answer
25 views

finding transition matrix and probability of a gambler's ruin [closed]

A gambler has \$2. At each play of a game,he loses \$1 with a probability $q$ but wins \$1 with probability $p$. He stops playing if he loses \$2 or wins \$4. i)What is the transition matrix $P$ of ...
1
vote
0answers
45 views

Expected time to failure

A machine needs two types of components in order to function. We have a stockpile of $n$ type-$1$ components and $m$ type-$2$ components. Type-$1$ components last for an exponential time with ...
0
votes
0answers
13 views

4th order correlations of a delta-correlated random process

Say I have a complex random variable A(z) that is $\delta$-correlated, i.e. I have: $ \begin{align}\langle A(z) \rangle &= 0 \\ \langle A(z) A^*(z') \rangle &= \delta(z-z') \end{align} $ ...